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Lattice QCD Method To Study Proton Radius Puzzle

T0 review · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A lattice QCD formula for the proton electric form factor is derived including a confinement boundary term, but no numerical calculation of the proton radius is performed.

arxiv 1908.01586 v1 pith:74LH7LLB submitted 2019-08-01 physics.gen-ph

Gouranga C Nayak This is my paper · ORCID
classification physics.gen-ph
keywords protonradiusmethodchargelatticecalculatedfactorform
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The proton radius puzzle is the disagreement between measurements of the proton's size made with ordinary hydrogen, electron scattering, and muonic hydrogen. The size is not a simple number because the proton is made of quarks and gluons; it is defined by the electric form factor, which describes how charge is distributed inside. Lattice QCD is the standard numerical method for calculating such non-perturbative quantities from the theory of strong interactions.

This paper writes down a formula for the proton electric form factor using lattice QCD. The derivation follows the usual procedure: compute two-point and three-point correlation functions on a lattice, take ratios to isolate the proton matrix element of the electromagnetic current, and then convert that matrix element into the electric form factor. The unusual ingredient is a 'boundary surface term' added to the proton energy, taken from the author's earlier papers, which is supposed to account for quark confinement inside a finite-size proton.

The paper does not actually run a lattice simulation or produce a number for the proton radius. Several steps are questionable. The boundary term is not independently justified, and the kinematic factor connecting the matrix element to the form factor appears to have a mass-dependent error. The formula is also written with indefinite integrations that are not spelled out in a way that a lattice group could directly implement.

Extended reading notes

Core claim

The paper claims that eq. (29) is 'the non-perturbative formula of the electric form factor G_E(Q^2) of the proton derived from the first principle in QCD', and that the charge radius R_P follows from eq. (1). If correct, this would be a first-principles lattice QCD method to compute the proton radius including a confinement boundary term.

Load-bearing premise

The derivation assumes that the non-zero boundary surface term E_BS(t) defined in eq. (11) and taken from the author's prior work (ref [14]) correctly accounts for quark confinement in the proton. This term enters the normalization factors in eqs. (27) and (28) and therefore appears in the final form factor formula (29). If E_BS(t) vanishes or has a different form, the claimed formula is not established.

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Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central formula rests on the standard lattice ratio method, which is well established, but the boundary-surface-term correction is an unverified input from the author's own prior papers. No fitted numerical parameters appear, since no lattice data are presented.

assumptions (3)
  • ad hoc to paper The non-zero boundary surface term E_BS(t) in QCD due to confinement exists and is given by eq. (11), taken from the author's prior work (ref [14]).
    This is the author's prior claim; it is not derived in this paper and lacks independent verification.
  • domain assumption The spectral decomposition and large-time ground-state dominance in eqs. (8)-(9) and (21)-(24) hold with indefinite integrals over Euclidean time.
    The use of indefinite integrals rather than fixed time separations is non-standard and not justified here.
  • domain assumption The relation between the proton matrix element and G_E given in eq. (26) is correct.
    The kinematic prefactor is stated without derivation and appears to differ from the standard relation.
invented entities (1)
  • Non-zero boundary surface term E_BS(t) in QCD due to confinement
    purpose: To modify the energy extracted from the two-point function, eqs. (27)-(28), and hence the normalization in the form factor ratio.
    No independent validation is offered; it is imported from the author's own previous papers.

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Pith. "Pith review of Lattice QCD Method To Study Proton Radius Puzzle." pith.science (2026). https://pith.science/paper/74LH7LLB

@misc{pith2026190801586,
  author       = {Pith},
  title        = {Pith review of: Lattice QCD Method To Study Proton Radius Puzzle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/74LH7LLB}},
  note         = {Machine review of arXiv:1908.01586}
}
read the original abstract

Recently there has been disagreement between various experiments about the value of the proton radius which is known as the proton radius puzzle. Since the proton is not a point particle the charge radius of the proton depends on the charge distribution (the form factor) of the partons inside the proton. Since this form factor is a non-perturbative quantity in QCD it cannot be calculated by using the perturbative QCD (pQCD) method but it can be calculated by using the lattice QCD method. In this paper we formulate the lattice QCD method to study the charge radius of the proton. We derive the non-perturbative formula of the charge radius of the proton from the first principle in QCD which can be calculated by using the lattice QCD method.

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Works this paper leans on

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